In this section, we develop the DFPV algorithm, which learns adaptive features modeled by neural nets using a technique similar to [1]}. As in Mastouri and Zhu et al. [2]}, we assume that we do not necessarily have access to observations from the joint distribution of \((A,Y,Z,W)\) . Instead, we are given \(m\)  observations of \((A,Z,W)\)  for Stage 1 and \(n\)  observations of \((A,Z,Y)\)  for Stage 2. We denote the stage 1 observations by \((a_i, z_i, w_i)\)  and the stage 2 observations by \((\tilde{a}_i, \tilde{z}_i, \tilde{y}_i)\) . If observations of \((A,Y,Z,W)\)  are given for both stages, we can evaluate the out-of-sample loss of Stage 1 using Stage 2 data and vice versa, and these losses can be used for hyper-parameter tuning of \(\lambda _1,\lambda _2\)  (Appendix ). We first introduce two-stage regression with adaptive feature maps, and then describe the detailed learning procedure of DFPV.
